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Physicists originating statistical mechanics asserted that the path in phase space of the state of the system will visit every point on the surface of constant energy. Bolzmann's ergodic hypothesis, as formulated by EhresmanEhrenfest, 1911, states exactly that. Later mathematicians found it to be in error for topological reasons: a differentiable curve cannot cover a surface of dimension higher than 1. You can then go on to study this...the curve is dense in the surface, and there are "ergodic" conditions much stronger than that which are (presumably) obeyed by physical systems.

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The ergodic hypothesis, is that what it was called?

Physicists originating statistical mechanics asserted that the path in phase space of the state of the system will visit every point on the surface of constant energy. Bolzmann's ergodic hypothesis, as formulated by Ehresman, 1911, states exactly that. Later mathematicians found it to be in error for topological reasons: a differentiable curve cannot cover a surface of dimension higher than 1. You can then go on to study this...the curve is dense in the surface, and there are "ergodic" conditions much stronger than that which are (presumably) obeyed by physical systems.